MMRCDISCUSSION PAPER SERIES

No. 484

Optimal Room Charge and Expected Sales under
Discrete Choice Models with Limited Capacity
Taiga Saito
Financial Research Center at Financial Services Agency, Government of Japan
Akihiko Takahashi
Graduate School of Economics, The University of Tokyo
Hiroshi Tsuda
Department of Mathematical Sciences, Doshisha University
March 2016
Revised June 2016

東京大学ものづくり経営研究センター
Manufacturing Management Research Center (MMRC)
Discussion papers are in draft form distributed for purposes of comment and discussion. Contact the author for
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http://merc.e.u-tokyo.ac.jp/mmrc/dp/index.html

Optimal Room Charge and Expected Sales under
Discrete Choice Models with Limited Capacity ∗†Taiga Saito ‡, Akihiko Takahashi §, Hiroshi Tsuda ¶June 20, 2016

Abstract
In this paper, we introduce a model that incorporates features of the fully transparent hotel booking systems and enables estimates of hotel choice probabilities ina group based on the room charges. Firstly, we extract necessary information for
the estimation from big data of online booking for major four hotels near Kyoto
station. 1 Then, we consider a nested logit model as well as a multinomial logit
model for the choice behavior of the customers, where the number of rooms available
for booking for each hotel are possibly limited. In addition, we apply the modelto an optimal room charge problem for a hotel that aims to maximize its expectedsales of a certain room type in the transparent online booking systems. We show
numerical examples of the maximization problem using the data of the four hotelsof November 2012 which is a high season in Kyoto city. This model is useful in that
hotel managers as well as hotel investors, such as hotel REITs and hotel funds, areable to predict the potential sales increase of hotels from online booking data andmake use of the result as a tool for investment decisions.

Keywords: Hotels in Kyoto, Revenue management, Online booking, Discrete choice
model

1

Introduction

Hotel revenue management is an important issue not only for hotel managers, but also
for hotel REITs and tourism funds who invest in hotels and engage in the management
in order to increase profitability from the properties. By the revenue management, they∗Forthcoming in International Journal of Hospitality Management.

†

All the contents expressed in this research are solely those of the authors and do not represent any
views or opinions of any institutions. The authors are not responsible or liable in any manner for any
losses and/or damages caused by the use of any contents in this research.
‡
Financial Research Center at Financial Services Agency, Government of Japan. 3-2-1 Kasumigaseki,
Chiyoda-ku, Tokyo 100-8967, Japan. saitotaiga@hotmail.com§
Graduate School of Economics, The University of Tokyo.
¶
Department of Mathematical Sciences, Doshisha University, Visiting Professor at the Institute of
Statistical Mathematics.
1
The data were provided by National Institute of Informatics.

1

are able to know how they operate the hotels in order to increase the revenues. For theinvestors in hotels, by knowing the sizes of potential sales increase, the revenue management can also be used as an investment decision making tool. Taking this into account, weconsider maximization of expected sales of a certain room type of hotels in a group in thesame area by solving for optimal room charges. The model incorporates random choiceof hotels by customers who visit an online hotel booking website. Then we use big dataof online booking for major four hotels near Kyoto station for the estimation, which werecollected from a Japanese booking website by National Institute of Informatics. They
include hotel names, accommodation plans with room types, remaining numbers of theplans available for booking, and prices of the plans for fourteen days booking periods priorto check-in dates. They are meaningful since those include more detailed information than
the financial statements which are usually undisclosed. We first extract necessary information from the data under some suitable assumptions so that it can be used to estimate
the parameters of the model.
For literatures on online hotel booking systems, Wang et al. [27] investigates relations
between the quality of hotel websites and customers’ online booking intentions. Noone
and Mattila [25] studies how the two ways of rate presentation for multiple-day stays,
the blended and non-blended presentations of best available rates, aﬀect customers’ willingness to book. Liu and Zhang [16] examines factors for travelers to choose an online
booking channel among hotel and online travel agencies websites. Casaló et al. [7] studies
eﬀects of online hotel ratings by online travel communities, such as TripAdvisor, on booking behaviors of customers. Ladhari and Michaud [15] investigates influences of online
word of mouth in social network services, such as Facebook, on the choice of a hotel.
Abrate et al. [1] examines dynamic pricing strategies of 1,000 European hotels. Viglia etal. [29] analyses the impact of hotel price sequences on customers’ reference price which
is used to evaluate market prices.
For applications of discrete choice models to tourism research, Masiero et al. [17],[20], Nicolau and Masiero [24], and Masiero and Nicolau [18] use the mixed logit modelfor the analysis of choice behaviors of hotel customers. Masiero et al. [17] investigatescustomers’ willingness to pay for hotel room attributes within a single hotel property.
Masiero et al. [20] examines an asymmetric preference for hotel room choices based onprospect theory. Masiero and Nicolau [18] analyses the determinants of individual pricesensitivities to tourism activities. Nicolau and Masiero [24] observes the eﬀect of individual
price sensitivities to tourism activities on on-site expenditures. Moreover, Viglia et al. [28]investigates impact of customer reviews on preference for hotel choice by a rank conjoint
experiment. Masiero et al. [19] examines determinant factors of tourist expenditure foraccommodation by a quantile regression.
Also, there are some other literatures on hotel revenue management by quantitative
approaches. Quantitative revenue management has been studied mostly on online booking
systems of airline tickets. For instance, Kimes [13], Weatherford and Bodily [26] summarize the methodologies. For hotels, Bitran and Mondschein [6], Badinelli [4] consider
sales maximization of a single hotel. In particular, Bitran and Mondschein [6] takes into
consideration the case of multi-day stays, and Badinelli [4] investigates dynamic room
pricing where the optimal policy depends on the vacancy and the remaining days before
the check-in date. Anderson and Xie [3] is the first study that deals with the sales maximization of hotels in a group, taking into account choice behaviors of customers in online
2

booking systems. Specifically, Anderson and Xie [3] studies on opaque booking systems
where the name of a hotel that customers try to book is concealed until the booking isdone, assuming the nested-logit model for the customer choices among hotels in diﬀerent
areas and price ranges. For qualitative analysis on practices of hotel revenue management,
see Baker and Collier [5], Donaghy et al. [8], Hanks et al. [10], Kims [14], Lieberman [23].
While Anderson and Xie [3] covers the opaque booking systems which are popularin the United States, the fully transparent booking systems, where customers consider
booking by knowing names of the hotels, are common in Japan. This is because customers
care for names and reviews of the hotels, since the quality of the services are dissimilareven among the same rank of hotels and some hotels diﬀerentiate themselves from therivals by selling variety of accommodation plans which include options such as tickets forsightseeing facilities, recreation, and meals.
In the transparent booking systems, particularly in high seasons, as customers are
only able to choose hotels oﬀering available rooms, the limitation of the number of rooms
that the rival hotels can oﬀer is an important factor to be considered in modeling. On
the other hand, in the opaque booking systems, choice categories, a pair of ratings andareas, are not exhausted as long as some of the hotels in the categories provide rooms.
Hence we model the transparent booking systems, taking into account the limitation of
the available number of rooms for the rival hotels. In the model, fully-occupied hotels
are excluded from the choice alternatives. We note that our model contains so called awaterfall structure as in collateralized debt obligations in finance. (See Gibson [9], Hulland White [12] for details.)
Moreover, in contrast to the opaque booking systems where the daily changing pricing
is a key to maximizing the sales, consistent pricing is important in the transparent booking
systems. As Anderson and Xie [3] points out, the frequent price changes and discounts
in a booking period in the fully transparent booking systems may lose loyal customersof the hotel who book in advance thorough the direct selling channel at regular highprices. Therefore, our work aims to obtain one optimal room charge, which is unchanged
through a booking period in the transparent booking systems, while Anderson and Xie [3]deals with daily pricing in the opaque booking systems in the United States where hotels
maximize their profits by selling out their rooms by discounting in several days before thecheck-in date.
The organization of the paper is as follows. Section 2 introduces the model thatreflects choice behavior of customers in the transparent online booking systems. The
model assumes a Poisson process for the frequency of visiting of customers and a nested
logit model, as well as a multinomial logit model, with limited number of available rooms
for hotels. An algorithm to calculate the expected sales under the model is also shown inthis section. Section 3 provides numerical examples of the optimal room charge. Finally,
Section 4 concludes. Appendices provide properties on the nested logit model in Section
2.1, and proofs of the theorem on existence of an optimal room charge in Section 2.2 andthe lemma in Section 3.2.

32

Model and Estimation

2.1

Model Specification

In this subsection, we introduce the model that describe booking activities of customers
in an online system who make a reservation for a certain type of room in a group of hotels.
Let {1, 2, . . . , M } be check-in dates. First fix a check-in date m, 1 ≤ m ≤ M . Let [0, T ]be a booking period for the check-in date, where 0 is the start date of the period and T
is a check-in date. We fix the check-in date. Let t ∈ [0, T ] be a booking date. We assume
that the total number of rooms booked during the period for the group for the check-in(m)date follows a Poisson process {Nt }0≤t≤T with intensity λ(m) . We further assume thefollowing.
• Fix the number of hotels in the group. Let L ∈ N be the number of hotels and wename the hotels from hotel 1 to hotel L.
• Let Rti , (1 ≤ i ≤ L) be the number of rooms of hotel i booked until time t thatsatisfiesL∑(m)

Rti = Nt

,

(1)

.

(2)

i=1
(m)

0 ≤ Rti ≤ qi(m)

Here qi ∈ N ∪ {∞} is the maximum number of rooms available for booking forhotel i (0 ≤ i ≤ L).
• Let γ ∈ Γ := ΠLi=1 {0, 1} be the state of full occupancy. That is, 0 for the i-thcomponent of γ means that there is no available room for the hotel i, otherwise thehotel is available for booking.
(m)γ

(m)γ

(0 ≤ pi≤ 1, 0 ≤ i ≤ L) be the choice probability of hotel i by a
• Let picustomer under the occupancy state γ, which satisfies
L
∑
(m)γ
pi
= 1.

(3)

i=1
(m)

(m)

• Let xi be the room charge of hotel i for the check-in date m and Vi(m)
R, (i = 1, 2, . . . , L) be linear functions of xi defined by
(m)

Vi

(m)

= αi + βi xi+ δi y (m) , (i = 1, . . . , L),
βi < 0.

: R →

(4)
(5)

where y (m) is a dummy variable that takes 1 if the check-in date m is a day before
a holiday or 0 otherwise.
(m)

We note that in the stochastic utility maximization theory as in McFadden [21], Vi(m)corresponds to the deterministic term of the random utility Ui of a customer for choosing
4

(m)

(m)

(m)

(m)

(m)

> Uj

hotel i with decomposition Ui = Vi +ϵi , where ϵi is the random term of the utility.
In the theory, since a customer choose hotel i with the highest utility against alternatives,(m)(m)
the choice probability for hotel i which depends on the distribution of (ϵ1 , . . . , ϵL ) is
given asP(Ui(m), i ̸= j).

(6)

In particular, Theorem 1 in McFadden [21] provides so called the generalized extreme value
model, a class of probabilistic choice models, which is consistent with utility maximization.
Hereafter, as choice probabilities we take a nested logit model (e.g. Section 9.3.5 inAmemiya [2]), which is originally developed by McFadden [21] as an example of generalized
extreme value models. (e.g. Section 5 in McFadden [21], Section 5.15 in McFadden [22])Assumption 1. (Choice probability) Let C1 , . . . , Cn be disjoint subsets of {1, . . . , L} satisfyingn∪Ck = {1, . . . , L}, Ck ∩ Cl = ∅.

(7)

k=1

Let ki ∈ {1, . . . , n} be the index such that i ∈ Cki and {j ∈ Cki |γj = 1} ̸= ∅. The choiceprobability of hotel i under the occupancy state γ ∈ Γ is given by
(m)γ

pi

{

=∑

∑

V

(m)

V

j
νki
j∈Ck exp( νk )1{γj =1} }

n
k=1 {

∑

i

i

(m)

Vj
j∈Ck exp( νk)1{γj =1} }νk

·∑

(m)

exp( νi k )1{γi =1}
i

V

(m)

(i = 1, . . . , L), (8)jj∈Ck exp( νk )1{γj =1}
i

i

(m)γ

where ν1 , . . . , νn are constants taking values 0 < ν1 , . . . , νn ≤ 1. We set pi{j ∈ Cki |γj = 1} = ∅.

= 0 whenWe notice that these choice probabilities correspond to the following joint distribution(m)(m)
function of the random terms of utilities, (ϵ1 , . . . , ϵL ) (e.g. Section 5 of McFadden[21]): for all w1 , . . . , wL ∈ R,
F (w1 , . . . , wL ) = exp (−G (exp(−w1 ), . . . , exp(−wL ))) ,

(9)

n
∑
∑ −ν
G(z1 , . . . , zL ) =(zj k 1{γj =1} )νk .

(10)

k=1 j∈Ck

Particularly, when ν1 = · · · = νn = 1, the choice probabilities agree with the ones in a
multinomial logit model:
(m)

(m)γ

pi

exp(Vi )1{γi =1}= ∑L, (i = 1, . . . , L).
(m)
exp(V
)1
{γj =1}
j
j=1

(11)

Remark 1. The parameters 1 − νk can be interpreted as the degrees of the similarity
within a set Ck . (For instance, see Section 5 in McFadden [21] and Section 5.15 in
McFadden [22] for the details.): when νk = 1, the choices in the set are dissimilar,
5and as νk decreases to 0, the choices in Ck become similar. More precisely, as in wellknown
properties of a generalized
mean (or a power mean), when νk goes to 0, the term
}νk
{
(m)
∑
Vj
in (8) is nondecreasing on 0 < νk ≤ 1, and converges toj∈Ck exp( νk )1{γj =1}(m)maxj∈C̃k exp(Vj

):
{
lim

νk →0

∑

(m)

Vj
exp(
)1{γj =1}
νk
j∈C

}νk
(m)

= max exp(Vj

),

j∈C̃k

k

where C̃k = {j ∈ Ck |γj = 1} ̸= ∅. For convenience, we give the proofs of those properties
in Appendix A.

2.2

Expected Sales of Hotel iIn this subsection, we observe how the expected sales of hotel i is calculated. Let Sk :=Πkj=1 {1, 2, . . . , L} be a set of booking orders when there are k bookings in total until
the check-in date. For each scenario (i1 , . . . , ik ) ∈ Sk , the corresponding scenario of the
(m)1
(m)2
(m)k
occupancy state (γ(i1 ,...,ik ) , γ(i1 ,...,ik ) , . . . , γ(i1 ,...,ik ) ) ∈ Πkj=1 Γ is defined as follows.
(m)j

(m)j

(m)j

γ(i1 ,...,ik ) := (γ1,(i1 ,...,ik ) , . . . , γL,(i1 ,...,ik ) ), (j = 1, . . . , k){∑(m)
1, if j−1,(m)jl=1 1{il =i} < qi
γi,(i1 ,...,ik ) :=0, otherwise.
(m)j

(12)
(13)
(m)j

Hereafter, for the notational simplicity, we abbreviate γ(i1 ,...,ik ) and γi,(i1 ,...,ik ) as γ(ij 1 ,...,ik )jand γi,(i, respectively. We assume that the conditional probability of the occurrence
1 ,...,ik )(m)of the scenario (i1 , . . . , ik ) ∈ Sk under NT(m)P({(i1 , . . . , ik )}|NT= k is given by
1(m)γ(i,...,i )= k) = pi11kk(m)γ(i,...,i ). . . pik

1

k

.

(14)

This has the following interpretations. For check-in date m, when the total number(m)of booking until the check-in date determined by the Poisson random variable NT isl(m)γ(i,...,i )1kk, the choice probability pilat the l-th booking (1 ≤ l ≤ k) only dependslon the occupancy state γ(i1 ,...,ik ) . Given the occupancy state, customers choose a hotel
irrespectively of the choices previously made. We notice that this mechanism where thenumber of alternatives decreases as they are filled up is analogous to a waterfall structure,a rule of prioritized cash payments in collateralized debt obligations. (See Gibson [9], Hulland White [12] for details.)
Then, the conditional expectation of the number of rooms booked for hotel i under(m)
NT = k is(m)

E[RTi |NT= k] =∑

1(m)γ(i,...,i )ai(i1 ,...,ik ) pi1(i1 ,...,ik )∈Sk

6

1

k

k
(m)γ(i,...,i ). . . pi k

1

k

,

(15)

whereai(i1 ,...,ik ) =k∑1{il =i} .

(16)

l=1

Therefore, the expectation of the sales of hotel i at the check-in date T is calculated
as follows.
(m)

(m)

E[xi RTi ] = xi E[RTi ]
∞
∑
(m)
(m)
(m)
= xi
E[RTi |NT = k]P(NT = k)
(m)

= xi

k=0
∞
∑

(m)

E[RTi |NT= k] exp(−λ(m) T )

k=0
∞
∑
(m)
= xi

∑

1
(m)γ(i,...,i )ai(i1 ,...,ik ) pi1

1k(λ(m) T )kk!
k(m)γ(i,...,i ). . . pi k1kexp(−λ(m) T )k=0 (i1 ,...,ik )∈Sk

(λ(m) T )k
.
k!
(17)

(m)

In the nested logit model, the expected sales attains its maximum at some xi
as in the following theorem. We provide the proof in Appendix B.

∈ [0, ∞)(m)

Theorem 1. Suppose that there exists j ̸= i such that qj = ∞. Then, in the nested(m)logit model (8), the expected sales (17) as a function of xi has its maximum point in[0, ∞).
(m)

Note that this assumption that there exists j ̸= i such that qj = ∞ indicates that
there is at least one hotel other than hotel i, which has enough capacity and can neverbe fully-occupied. In the case where there is no such hotel, if all the other hotels are
fully-booked, the choice probability of hotel i becomes 1 regardless of the room charge of(m)hotel i. In such a case, the expected sales is increasing with respect to xi and does nothave a maximum point.

3

Numerical Example

In this section, we show numerical examples of the maximization of the expected sales by
the multinomial logit model and the nested logit model with limited capacity introducedin Section 2. We use empirical online booking data of major four hotels near Kyoto station
which were oﬀered by National Institute of Informatics and examine the maximization
problem of the four hotels. We consider choice behaviors of customers who seek to stayat a hotel very near Kyoto station, where we assume that they choose a hotel among
the major four hotels close to the station. In Kyoto, the station neighborhood is amajor accommodation area and the four hotels are luxury full-service hotels. Hence, it
is reasonable to consider that there exists a group of customers who prefer to choose adecent high-class full-service hotel among the four in this area. The validity of selecting
the four hotels for the choices in the analysis will be discussed in more detail in Section
7

3.3. For the room type customers choose in the model, we adopt nonsmoking standard
twin room, a common room type for tourists. We first estimate parameters of the models
with limited capacity by the maximum likelihood method on the data of November in
2012, which is the busiest season in Kyoto when tourists enjoy watching the fall foliage.
Especially, since the latter half of November is the peak season of the fall foliage andthe hotels set high room charges regardless of whether a check-in date is a day before aholiday, we treat the last two weeks in November 2012 as a day before a holiday. We
also note that the Japanese yen (JPY) had been strong against other currencies in 2012,
and the year was right before the number of tourists from overseas started to rise and thehotels were constantly occupied by these tourists. Hence, in order to exclude the large
eﬀect of overseas travelers on booking, we use the data of this year. Then we consider
maximization of the expected sales of the hotels for two check-in dates, a normal weekday
and a day before a holiday including the last half of November.
In the following of this section, after Section 3.1 explains details of the data set usedin the numerical example, Section 3.2 provides estimation results of the parameters in themultinomial logit model and the nested logit model, and Section 3.3 discusses validity ofthe data set in the analysis. Finally, Section 3.4 shows the optimal room charges and the
expected sales for the two types of check-in dates calculated with the estimated nested
logit model.

3.1

Data Set

We use online booking data of major four hotels near Kyoto station, which were collectedfrom a Japanese booking website by National Institute of Informatics. The four hotels
are named as Hotel A, Hotel B, Hotel C and Hotel D in this paper. The original data
include available numbers for booking for all accommodation plans and their prices foreach booking date of all check-in dates.
Note that the original data contain no information on booked numbers either per room
type or per accommodation plan. Moreover, as the available number for booking of an
accommodation plan changes in tandem with other plans that oﬀer the same room type,
we do not know which plan was sold or how many rooms were booked for the room typewhen the available number for booking decreases. Thus, we have to estimate the booked
number on a specific room type in some way. In addition, since a room type is overlapped
among various accommodation plans and their prices are diﬀerent, in order to comply
with the model, we have to define a representative price of the room type for a check-in
date.
Considering these points, we initially convert the original data to a data set used forour analysis in the following way. For the number of rooms booked for a specific room
type, we first calculate changes from the previous day in the available number of booking
for each accommodation plan. If the change is a decrease, we assume that this number ofplans were booked. If the available number of booking is unchanged, we assume that noplan was booked, and if the change is an increase, this number of plans were just suppliedby the hotel. Then we take a maximum of these numbers among all accommodation plansthat oﬀer this room type. We assume the maximum as the number of rooms booked forthis room type.
Next, since in practice, hotels occasionally change their room charges in some degree8during a booking period, and there do not exist single prices as in the model for the checkin dates, we take an average of the prices over the booking period for each accommodation
plan. Then, we take a minimum of the average prices among all the accommodation plansthat oﬀer this room type, that is, nonsmoking standard twin room, in order to find theaverage price with no or minimal options. We assume the minimum as the representative(m)room charge over the booking period for the room type. This will be denoted by xi inthe following.
The data set includes booking information of the 30 check-in days in November 2012
for the standard nonsmoking twin room type of the four hotels. For each check-in date,
the numbers of rooms booked for 14 booking dates ranging from the check-in date to 13
days prior to the check-in date, and the representative room charges over the booking
period are available. Table 1 shows an example of the booking data for the check-in dateof a weekday in November 2012. The right side of the table shows the numbers of rooms
sold for this room type for each hotel on each booking date. ”Full” in Table 1 indicatesthat no room was available for booking as the rooms were sold out before the date.
check-in date
-1 day-2 days-3 days-4 days-5 days-6 days-7 days-8 days-9 days-10 days-11 days-12 days-13 daysnumber of rooms booked

Hotel A
3
0
2
1
0
0
0
0
1
1
1
3
0
0
12

Hotel B
Full
Full
Full
Full
3
0
2
0
2
4
0
0
2
0
13

Hotel C
0
1
0
11
0
0
0
0
0
0
18
2
0
0
32

Hotel D
5
2
0
2
0
0
1
0
4
5
1
0
8
0
28

Table 1: Estimated Numbers of Rooms Booked for Check-In Date, a weekday in November
2012 with Representative Room Charges 11,985, 11,000, 19,938, 18,000 in JPY for HotelA, Hotel B, Hotel C, and Hotel D
Table 2 summarizes the estimated data set. Out of a total of 1,680 (= 30days ×
14days × 4hotels) booking days for the four hotels, 1,273 days were available for booking,and the rest of the days were unavailable due to the full occupancy. In the available
booking days, 1,856 rooms were booked in total. Note that the room charges in Table 2
and prices hereafter are all expressed in JPY.

9

maximum representative room charge per check-in date
minimum representative room charge per check-in dateaverage room charge for whole check-in dates
maximum representative room charge per check-in date, a weekday
minimum representative room charge per check-in date, a weekday
average representative room charge per check-in date, a weekday
maximum representative room charge per check-in date, a day before a holiday
minimum representative room charge per check-in date, a day before a holiday
average representative room charge per check-in date, a day before a holiday
maximum number of rooms booked per check-in date
minimum number of rooms booked per check-in dateaverage number of rooms booked for per check-in datetotal number of rooms booked for whole check-in dates
maximum number of available booking days per check-in date
minimum number of available booking days per check-in dateaverage number of available booking days per check-in datetotal number of available booking days for whole check-in dates
maximum number of accommodation plans per check-in date
minimum number of accommodation plans per check-in dateaverage number of accommodation plans per check-in dateHotel A
24,252
11,292
17,502
16,886
11,292
13,85824,252
15,70021,175
61
0
15.33
460
14
0
9.93
298
36
32
34.14

Hotel B
18,5009,571
15,071
18,5009,571
13,14918,085
15,236
17,321
15
0
5.03
151
14
0
8.07
242
11
11
11.00

Hotel C32,32918,24627,24627,293
18,24622,02032,32926,27730,387
37
0
10.27
308
14
0
10.57
317
5
5
5.00

Hotel D
30,03015,286
21,17022,286
15,286
19,61230,03018,857
22,122
72
0
31.23
937
14
12
13.87
416
38
2
20.93

Table 2: Estimated Data Summary, November 2012.

3.2

Estimation

In this subsection, we provide estimation results of the multinomial logit model and thenested logit model with the data described in the previous subsection.
First, the next lemma indicates that in the nested logit model as well as the multinomial logit model, the choice probability is reexpressed as follows, whose proof is given in
Appendix C.
Lemma 1. Let
(m)

(m)

Ṽ1

= β1 x1 ,(m)(m)
Ṽi = α̃i + βi xi + δ˜i y (m) (i = 2, . . . , L),

(18)
(19)

where α̃i = αi − α1 , δ˜i = δi − δ1 .
(m)γThen, piin the nested logit model (8) is rewritten as follows.
(m)γ

pi

{

=∑

∑

Ṽ

(m)

Ṽ

j
νki
j∈Ck exp( νk )1{γj =1} }(m)exp( νi k )1{γi =1}·∑, (i = 1, . . . , L).
(m)
(m)
∑
Ṽj
Ṽj
n
ν
k
k=1 {
j∈Ck exp( νk )1{γj =1} }j∈Ck exp( νk )1{γj =1}
i

i

i

i

i

(20)
In particular, when νk1 , . . . , νkL = 1 (the multinomial logit model (11)),
(m)

(m)γ

pi

exp(Ṽi )1{γi =1}= ∑L(i = 1, . . . , L).
(m)
exp(
Ṽ
)1
{γi =1}
j
j=1

(21)

Then, Table 3,4,5,6 show the estimation results of the coeﬃcients of the explanatory
variables in the nested logit model (20) and the multinomial logit model (21).

10Estimate Std. Error
α̃2 , Hotel B:(intercept)-0.7151
0.5097α̃3 , Hotel C:(intercept)
0.14550.5506α̃4 , Hotel D:(intercept)
0.82010.3450δ̃2 , Hotel B:holiday-1.03460.3104δ̃3 , Hotel C:holiday-0.6338
0.3169δ̃4 , Hotel D:holiday-0.5633
0.2055β1 , Hotel A:price-0.000125 0.0000330
β2 , Hotel B:price-0.000122 0.0000457
β3 , Hotel C:price-0.000099 0.0000267
β4 , Hotel D:price-0.000118 0.0000268
ν1 , iv.Hotel A,Hotel D
0.670
0.179
ν2 , iv.Hotel B,Hotel C
0.594
0.189

t-value p-value-1.4029 0.161
0.26420.792
2.3771
0.017-3.3331 0.001-2.0003 0.045-2.7416 0.006-3.7733 0.000-2.6690.008-3.6976 0.000-4.3953 0.000
3.74730.000
3.13890.002

Table 3: Estimation Results, Nested Logit Model, November 2012.

Log-Likelihood
McFadden R2
Likelihood ratio test

-1956.2
0.11652
χ = 515.91 (p.value ≤ 2.22e-16)

Table 4: Estimation Results (Tests), Nested Logit Model, November 2012.

α̃2 , Hotel B:(intercept)
α̃3 , Hotel C:(intercept)
α̃4 , Hotel D:(intercept)δ̃2 , Hotel B:holidayδ̃3 , Hotel C:holidayδ̃4 , Hotel D:holiday
β1 , Hotel A:price
β2 , Hotel B:price
β3 , Hotel C:price
β4 , Hotel D:price

Estimate Std. Error-0.55390.72070.40130.60281.10900.4651-1.1303
0.4127-0.6523
0.2956-0.73780.2404-0.000167 0.0000307-0.000185 0.0000570-0.000129 0.0000272-0.000157 0.0000214

t-value p-value-0.7686 0.442
0.66580.506
2.38460.017-2.7388 0.006-2.2064 0.027-3.0692 0.002-5.4509 0.000-3.2465 0.001-4.7452 0.000-7.3216 0.000

Table 5: Estimation Results, Multinomial Logit Model, November 2012.

Log-Likelihood
McFadden R2
Likelihood ratio test

-1959
0.115222
χ = 510.22, (p.value ≤ 2.22e-16)

Table 6: Estimation Results (Tests), Multinomial Logit Model, November 2012.
11

We estimate the parameters in the models by the maximum likelihood method withthe maximum likelihood functions of the following forms:(the nested logit model){ (m)γ}η(m,l,i)(m)(m) (m)
(m,l)
13
4
p
Π30
Π
Π
(x
,
.
.
.
,
x
,
y
;
α̃
,
α̃
,
α̃
,
β
,
.
.
.
,
β
,
δ̃
,
δ̃
,
δ̃
,
ν
,
ν
)
.
2
3
4
1
4 2 3 4 1 2
1
4
m=1 l=0 i=1i(22)(the multinomial logit model){ (m)γ}η(m,l,i)(m)(m) (m)
(m,l)
13
4
Π30
Π
Π
p
(x
,
.
.
.
,
x
,
y
;
α̃
,
α̃
,
α̃
,
β
,
.
.
.
,
β
,
δ̃
,
δ̃
,
δ̃
)
.
2
3
4
1
4 2 3 41
4m=1 l=0 i=1i(23)
In this estimation, we assume the following. Let hotel 1, hotel 2, hotel 3, hotel 4 be
Hotel A, Hotel B, Hotel C and Hotel D respectively. Let check-in date 1, . . . , check-in
date 30 be the 30 check-in dates in November 2012 in ascending order. For each checkin date, there are 14 days of the booking period ranging from the check-in date to the
13 days prior to the check-in date. We call the booking date, which is l days prior to
(m)
(m)
(m)
(m)
the check-in date (0 ≤ l ≤ 13), booking date l. Let x1 , x2 , x3 , x4 (1 ≤ m ≤ 30)
be the representative room charge defined in Section 3.1 of the nonsmoking standard
twin room of hotel 1, hotel 2, hotel 3, hotel 4 over the booking period for check-in
date m. Let γ(m,l) be the occupancy state of booking date l for the check-in date m,and η(m,l,i) be the number of rooms booked for hotel i on booking date l for check(m)γ

(m)

(m)

in date m. Also, pi (m,l) (x1 , . . . , x4 , y (m) ; α̃2 , α̃3 , α̃4 , β1 , . . . , β4 , δ̃2 , δ̃3 , δ̃4 , ν1 , ν2 ) and
(m)γ
(m)
(m)
pi (m,l) (x1 , . . . , x4 , y (m) ; α̃2 , α̃3 , α̃4 , β1 , . . . , β4 , δ̃2 , δ̃3 , δ̃4 ) represent the choice probabilities (20) and (21), respectively.
For the nested logit model, we assume that the alternatives are grouped into two nests:
Hotel A and Hotel D which oﬀer a large number of accommodation plans in online booking
systems, more than 30 plans at a maximum and more than 20 plans on average for theroom type in November 2012 as in Table 2 for example, and Hotel B and Hotel C whichsell several accommodation plans. Hence we set C1 = {1, 4}, C2 = {2, 3}. We remark that
we also estimated the nested logit model with the other possible nesting structures. In
any of these cases, the parameters were not estimated properly. The results were either
calculation failure of the maximum likelihood method or inappropriate parameters withνk greater than 1.
In Table 3 and 5, βi (i = 1, 2, 3, 4) show all negative signs. This agrees with theintuition that as the room charge of a hotel increases, the utility of customers for the
hotel decreases. The negative signs of the coeﬃcients of the holiday factor δ̃i (i = 2, 3, 4)
indicate that for check-in dates, each of which is a day before a holiday, an increase of
the utility for Hotel B,C,D from weekday check-in dates is less than that of Hotel A.
Moreover, the low absolute values of t-values for α̃2 , α̃3 in Table 3 and 5 indicate thatthe intercepts αi are not very diﬀerent between Hotel A and Hotel B or Hotel A andHotel C. However, α̃2 , α̃3 have a statistical significance when the explanatory variables
xi are centered. For example, if we shift xi by 16,020, which is the average of all eight(m)(m)
minimum representative room charges in Table 2, that is, if we set x̂i = xi − 16,020,12

(i = 1, 2, 3, 4) and estimate the parameters with
(m)

V̂1

(m)

= β̂1 x̂1 ,(m)(m)
V̂i = α̂i + β̂i x̂i + δ̂i y (m) (i = 2, 3, 4),

(24)
(25)

we observe that the t-values of all the intercepts in the nested logit model as well as themultinomial logit model improve, and the lowest absolute value of a t-value is 1.89 of α̂3
in the nested logit model. This implies that when the prices are centered at the level, theintercepts have a significance.
We examine the intercepts in the centered prices case in Section 3.3 by computing the
choice probabilities when all the prices are at the same levels.
Also, as stated in Remark 1 in Section 2.1, 1 − ν1 (∼ 0.33) and 1 − ν2 (∼ 0.41) can beinterpreted as measures of the similarity of the two alternatives in C1 and C2 , respectively.
Moreover, it is known that for the case of two alternatives within the same set Ck , 1 − νk2
is equivalent to the correlation between the two alternatives. (e.g. p.300 in Amemiya [2].)Hence, the correlations between two alternatives are around 0.55 and 0.65 within C1 andC2 , respectively.
Next, we conduct the Hausman test for the multinomial logit model in Hausman and
McFadden [11]. The Hausman test judges whether the IIA (Independence from Irrelevant
Alternatives) property holds in the estimated multinomial logit model. The test checks
consistency of the estimated parameters between the original model and the model withfewer choices which are part of the full choices. We observe that the null hypothesis that
the IIA property holds is rejected for subsets of choices excluding Hotel C or Hotel D.
Hence, in the following examples, we only consider the nested logit model.

3.3

Discussion on Validity of Data Set

In this subsection, we discuss validity of the data set used in this numerical example
and also address its limitations. We have selected full-service hotels in front of Kyoto
station for the choices. These four hotels are the only full-service hotels in this locationequipped with multiple restaurants, lounges, banquet rooms, conference halls for MICE
(Meetings, Incentives, Conventions, Exhibitions) and a wedding center. We note that
Kyoto city is designated as a global MICE strategic city by Japan Tourism Agency andtakes its tourism strategy aiming for the ripple eﬀects by the MICE tourists. On the other
hand, the other hotels in this location are either select-service hotels or limited-service
hotels, whose target customers are business travelers or leisure tourists putting a highvalue on costs while expecting minimal services from the hotels. Here, we mean ’in frontof Kyoto Station’ by within 500 meter distance of Kyoto station. In fact, the four hotels
are all located within two-minute walking distance of Kyoto station, which oﬀers great
convenience to tourists coming from other parts of Japan by the superexpress train orother countries via Tokyo. Hotel C is in the terminal building, Hotel A is adjacent to theeast side of the station, and Hotel B and Hotel D stand at the north or south front of thestation.
In other words, we suppose that the target customers in this numerical analysis are theones who look for a full-service hotel in front of Kyoto station, that is, aim to choose fromthe four hotels above. Among the four hotels, the price seems the primary characteristic
13

that diﬀerentiates the hotel from the others and aﬀects choice behaviors of customers.
Hence, we set the price as the explanatory variable along with the holiday factor.
We remark that given the same location as in front of Kyoto Station, other attributes
such as rating, customer reviews and capacity are inherent characteristics to each hotel,
which remain unchanged for the short period in this analysis. Their eﬀect on the random(m)utility is reflected in the intercept αi of Vi in (4). Even if these characteristics are added(m)
as new explanatory variables for Vi , their coeﬃcients are not reasonably estimated due
to multicolinearity of the variables.
One way to observe these eﬀects within our model is computing the choice probabilities
when Hotel A, Hotel B, Hotel C and Hotel D oﬀer the same prices. Although there issome impact from the price term, the choice probabilities, when the four room charges
are the same, are considered to reflect the eﬀect of the other characteristics on customers
choice behaviors. Table 7 and 8 show the result, where we observe that for both aweekday and a day before a holiday, in all the cases, the ranking of choice probabilities is
D > C > A > B. We note that if we focus on Hotel A, Hotel B and Hotel C, the rankingof the choice probabilities is the same as the order of ratings and customer reviews, which
are examined in detail below. As for the choice probability of Hotel D, it is the highest
in the rankings for both the weekday and the day before a holiday. Considering that
Hotel D has the largest capacity as observed in Table 9 below and oﬀers large number ofaccommodation plans as shown in Table 2, the capacity, promotion and advertisement arealso considered to be the characteristics that aﬀect the choice behavior of the customers.
Room charge Hotel A
10,00013%
15,00012%
20,00011%
25,00011%30,00010%Hotel B
5%
4%
4%
3%
3%

Hotel C
32%
34%
37%
40%
42%

Hotel D
50%
49%
48%
46%
45%

Table 7: Choice Probabilities of the Hotels for the same room charges, a weekday in
November 2012.
Room charge Hotel A
10,00027%
15,00025%
20,00023%
25,00022%
30,00020%

Hotel B
2%
2%
2%
2%
1%

Hotel C
28%
31%
33%
36%
38%

Hotel D
43%
43%
42%
41%
40%

Table 8: Choice Probabilities of the Hotels for the same room charges, a day before aholiday in November 2012.
Next, Table 9 shows the summary of ratings, customer reviews and capacity for thefour hotels. Although there is no oﬃcial rating agency for Japanese hotels in Japan,
Michelin Guide selects some excellent hotels in Kyoto and gives them its own rating. We
14

can also refer to the ratings by online booking websites such as Booking.com and Expedia,
though only the latest ratings are available. Then, we observe that Hotel C and HotelD earn 4 pavilions and 2 pavilions respectively in Michelin Guide 2012. The ranking of
the ratings is C > A = D > B in Booking.com and C > A > D = B in Expedia. We
refer to Rakuten Travel for customer reviews, since the customers in the data set are theones who booked through a Japanese booking website. We have aggregated the customer
reviews in Rakuten Travel made for stays in 2012 for the four hotels and taken average ofthe ratings in the reviews. Then, the ranking of the customer reviews is C > A > D > B.
As for capacity, the ranking of the total number of guest rooms is D > C > A > B.
These numbers are as of April 2016 and unchanged from 2012.
rating, Michelin Guide 2012
rating, Booking.com as of April 2016
rating, Expedia as of April 2016
customer reviews, Rakuten Travel as of 2012
total number of guest rooms

Hotel A
N/A
4
4.0
4.27
220

Hotel B
N/A
3
3.5
4.17
160

Hotel C
4
5
4.5
4.65
535

Hotel D
2
4
3.5
4.25
988

Table 9: Rating, Customer Reviews and Capacity for Hotel A, Hotel B, Hotel C andHotel D.
Finally, we discuss the limitations of the data set: First, the customer type may bepossibly diﬀerent from what we have assumed. For instance, some customers who only
care about the location may choose a hotel from a wider range of hotels in front of thestation regardless of their service type. Second, although we have used the data as ofNovember in 2012, this short period data do not reflect the change in the attributes otherthan the price. In order to analyze the eﬀect of these attributes on choice behaviors ofcustomers more thoroughly, the longer period data would be more appropriate. Third,
the accuracy of the parameters estimation depends on the data quality. We have made theassumptions as in Section 3.1 in order to estimate the number of rooms booked and thequantity of inventory from the original online booking data. If we have full informationon these items which each individual hotel has, the accuracy of the parameters estimation
improves. However, it is also the advantage of the data set that hotels can maximize their
expected sales by utilizing this data on the rival hotels observable at booking websites.

3.4

Estimated Optimal Room Charge and Expected Sales

In this subsection, we show the optimal room charges with the nested logit model in thecases of a normal weekday and a day before a holiday for the four hotels. Particularly, thisresult is obtained by the previously explained method with assumptions for constructionof the data set described in Section 3.1. We also provide examples of the equilibrium room
charges where every hotel maximizes its expected sales simultaneously. First, we show an
example of the optimal room charge for a weekday. Hereafter we abbreviate the suﬃx m
on the parameters for the notational simplicity. In addition to the estimated parameters of
the nested logit model in Section 3.2, we assume the following: λ = 6.07, x1 = 11,985, x2 =11,000, x3 = 19,938, x4 = 18,000, T = 14, q1 = ∞, q2 = 13, q3 = ∞, q4 = ∞. They are
estimated from the data on the check-in date of a weekday in November 2012, when 85
15

rooms were booked over the 14 booking dates, the representative room charges for thefour hotels were 11,985, 11,000, 19,938, 18,000 for Hotel A, Hotel B, Hotel C, and Hotel D
respectively, and 13 rooms were booked for Hotel B before it got fully occupied while theother hotels had some rooms available at the check-in dates. We set qi as the number ofrooms sold during the booking period for the hotels which got fully-occupied, otherwise
we set qi = ∞. Note that in reality, each hotel knows its own quantity of room-inventory
qi , but needs to predict the quantity of the other hotels. We estimated λ by the maximum
likelihood method for the Poisson distribution, which means that we set λ as the total
number of rooms booked for the four hotels divided by 14, the number of booking days forthe check-in date. Note that in order to enable the scenario by scenario computation of the
expected sales in (17), which includes summation of large number of scenarios with low
probabilities, we initially rescale the number of rooms of the hotels by considering 20 rooms
as one batch. Hence we use the parameter λ̃ = 0.304, q̃1 = ∞, q̃2 = 1, q̃3 = ∞, q̃4 = ∞qi
λwhere λ̃ = 20
, q̃i = [ 20] + 1, (i = 1, 2, 3, 4) in (17), where [·] stands for the Gauss symbol.
After the computation, the expected sales is obtained by multiplying the result by 20.

Figure 1: Relation between Estimated Expected Revenue and Room Charge (Individual
Maximization Case), a weekday in November 2012, Hotel A, Hotel B, Hotel C and HotelD.
Figure 1 shows the graphs of the expected sales as a function of the room charge forHotel A, Hotel B, Hotel C, and Hotel D for the check-in date of a weekday in November
2012. We observe that as the room charge increases, the expected sales initially increases,
and after the maximum point it turns to decreasing. The reason is that the higher room
charge outweighs the decline in the expected number of rooms booked at first, and viceversa after the maximum point. These graphs also indicate the levels of the optimal room
charges that maximize the expected sales of the hotels. (1) and (2) in Table 10 and 11
summarize the listed and optimal room charges, and their expected sales. We observe that
Hotel A, Hotel C and Hotel D provided higher room charge than the estimated optimallevel, while Hotel B oﬀered a room charge close to the optimal level for this check-in date.
We also find that each hotel potentially improves its estimated expected sales by setting
the room charge at the optimal level.

16Figure 2: Relation between Estimated Expected Revenue and Room Charge (EquilibriumCase), a weekday in November 2012, Hotel A, Hotel B, Hotel C and Hotel D.
Figure 2 illustrates the expected sales of the hotels, when the other hotels set theirroom charges at the equilibrium prices. Here the equilibrium prices are the room chargesof the hotels such that for each hotel, the price is at its optimal level given the otherthree hotels room charges at the equilibrium levels. In other words, they are the prices
with which all the hotels maximize their expected sales simultaneously. The equilibrium
room charge and the corresponding expected sales are (7,500, 137,873), (7,000, 47,112),(12,000, 250,380), (11,500, 383,706) for Hotel A, Hotel B, Hotel C, and Hotel D respectively. The equilibrium prices are obtained by iteration of the expected sales maximization
where the room charge is replaced by the optimal one at each step. In detail, given thelisted prices of the four hotels, we first obtain the optimal room charge of Hotel A. Then
we replace the room charge of Hotel A by this, and calculate the optimal room chargeof Hotel B. We repeat this maximization in the order of Hotel A, Hotel B, Hotel C, andHotel D until the set of the four prices converges. Note that we have obtained the sameequilibrium levels even if we switch the starting point of the iteration to Hotel B, Hotel Cor Hotel D in this example as well as in the following example in Table 12 and 13, whichshows the robustness of the obtained equilibrium prices at least around the individual
optimization levels. Figure 1 and 2 show that in the equilibrium case, the expected sales
are substantially reduced from the individual maximization. This implies that when all
the hotels optimize their room charge simultaneously, as a result of the price competition,
the expected sales fairly declined.
Table 10 and 11 summarize the optimal room charges and the expected sales for theindividual maximization and the equilibrium cases. Compared to the individual maximization case, the equilibrium room charges are all at lower levels for this check-in date.
It is also observed that the expected sales in the equilibrium case are lower than theindividual maximization case by 34% to 51%, and than the listed case by 29% to 51%.

17

(1) Listed(2) Individual Maximization(3) Equilibrium

Hotel A
11,9859,500
7,500

Hotel B
11,00010,0007,000

Hotel C
19,93814,00012,000Hotel D
18,00014,00011,500

Table 10: Room Charges for Listed, Individual Maximization, and Equilibrium, a weekday
in November 2012.

(1) Listed(2) Individual Maximization(3) Equilibrium

Hotel A
268,749
283,165
137,873

Hotel B
95,45496,743
47,112

Hotel C366,263415,515250,380

Hotel D537,706
584,968
383,706

Table 11: Estimated Expected Sales for Listed, Individual Maximization, and Equilibrium, a weekday in November 2012.
The second example provides the case of a day before a holiday for the check-in
date, when tourists stay for sightseeing for the weekend. Here we assume the following
parameters: λ = 3.93, x1 = 18,036, x2 = 17,771, x3 = 26,400, x4 = 20,000, T = 14, q1 =∞, q2 = 7, q3 = 29, q4 = ∞. Similarly to the previous example, they are estimated fromthe booking data on a day before a holiday in November 2012, when 55 rooms were booked
in total for the four hotels in 14 booking days, Hotel B and Hotel C became fully occupied
after 7 and 29 rooms were booked respectively, while the other hotels still had some rooms
available at the check-in date, and 18,036, 177,771, 26,400, 20,000 were the representativeroom charges for Hotel A, Hotel B, Hotel C and Hotel D. As in the first example, we
rescale the room number by regarding 20 rooms as one batch and multiply the number
to the computational result. Hence we use λ̃ = 0.196, q̃1 = ∞, q̃2 = 1, q̃3 = 2, q̃4 = ∞ forthe computation of (17).

Figure 3: Relation between Estimated Expected Revenue and Room Charge (Individual
Maximization Case), a day before a holiday in November 2012, Hotel A, Hotel B, Hotel
C and Hotel D.
18

Figure 4: Relation between Estimated Expected Revenue and Room Charge (EquilibriumCase), a day before a holiday in November 2012, Hotel A, Hotel B, Hotel C and Hotel D.
Figure 3 and 4 illustrate the relation between the expected sales and the room charge
for Hotel A, Hotel B, Hotel C, and Hotel D for both the individual maximization andthe equilibrium cases. Table 12 displays the optimal room charges in the cases of theindividual maximization and the equilibrium, as well as the listed room charges, andTable 13 shows their corresponding expected sales.

ListedIndividual Maximization
Equilibrium

Hotel A
18,036
11,0007,500

Hotel B
17,7718,000
6,000

Hotel C
26,40019,50014,000Hotel D
20,00018,00012,000

Table 12: Room Charges for Listed, Individual Maximization, and Equilibrium, a day
before a holiday in November 2012.

ListedIndividual Maximization
Equilibrium

Hotel A
190,464
272,844
110,506

Hotel B
22,277
46,682
11,662

Hotel C
320,845
363,768
186,397

Hotel D
610,295
618,496
293,878

Table 13: Estimated Expected Sales for Listed, Individual Maximization, and Equilibrium, a day before a holiday in November 2012.
The numerical results in the second example, show that the room charges in the equilibrium case are all lower than those in the individual maximization. Table 12 indicatesthat in the equilibrium situation where all the hotels aim to maximize their sales simultaneously, the room charges settle at low levels as a result of the price competition, which
also leads to the substantially lower expected sales as in Table 13. It is observed that the19expected sales in the equilibrium case are lower than those in the individual maximizationcase by 49% to 75%, and than in the listed case by 42% to 52%.
We can observe for both examples in Table 11 and 13 that expected sales with thelisted room charge are in between those with the individual maximization and equilibrium
prices. This may suggest that the hotels avoid lower expected sales caused by pricecompetition which arises from the individual maximization of the hotels. On the other
hand, the results also indicate that the hotels can increase their expected sales by setting
their room charges at their estimated optimal levels by the individual maximization aslong as the other hotels do not change their room charges to their optimal levels.

4

Conclusions and Future ResearchThis paper has analyzed online booking data of Kyoto, a city of international tourism
that has 17 World Heritage Sites, for the first time in the literatures of hotel revenue
management. The revenue management model used in this study reflects unique featuresof Japanese booking websites, fully transparent booking systems and limitation of thenumbers of room available for booking. Firstly, this study has applied a quantitative
revenue management model for estimates of choice probabilities of hotels by customers
in online booking systems, which depend on room charges and types of a check-in dateof hotels. The parameters in the model are estimated from a data set based on actual
online booking data of customers for major four hotels in Kyoto city with application of
econometric models, such as the multinomial and nested logit models. We have inferred
the actual numbers of rooms booked under some appropriate assumptions, since these are
not available in the original data set. This inference is meaningful because by extracting
the information on rival hotels from data openly available on the web and using the model,
hotels are able to predict their expected sales and optimal room charge. We have predicted
optimal room charges and expected sales of the hotels when the other hotels’ room charges
are fixed or the other hotels also simultaneously maximize their expected sales, which isclearly useful for hotel managers. Moreover, this model enables hotel investors, such as
hotel REIT and hotel funds to evaluate business value of hotels.
Finally, examining choice behaviors of customers in a longer period will be our future
research topic: Particularly, we consider the other characteristics that do not change ina short term as explanatory variables, and we also take a wider range of hotels as thecandidates for the choices.
Although we have used the data as of 2012, which was the year before the number ofoverseas tourists started to rise, investigating how the choice behaviors of customers havechanged by the increase of the overseas tourists resulting from JPY weakening, is also ournext research topic.

5

Acknowledgments

We are grateful to the editor-in-chief, Professor Abraham Pizam, the managing editor, Professor Manuel Rivera and two anonymous referees for their precious comments,
which substantially improve the previous version of this paper. We also thank Professor Naoyuki Yoshino, Professor Nobuo Takahashi, Professor Takashi Obinata, Professor
20

Masaki Yoneyama and Professor Seisho Sato for their valuable suggestions. We are grateful to National Institute of Informatics for oﬀering data used in this study and to Kotaro
Urakami for organizing the database. This research is supported by JSPS KAKENHI
Grant Numbers 25380389 and 26283019.

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22

A

Some properties of the equation (8) with respect
to νk

Proposition 1. Let C̃k = {j ∈ Ck |γj = 1}. Suppose that C̃k is nonempty. Then we have}νk{
( (m) )∑Vj(m)
1{γj =1}= max exp(Vj ).
lim
(26)
exp
νk →0
ν
j∈
C̃
k
k
j∈C
k

Moreover,

{

∑

(
exp

j∈Ck

(m)

Vj
νk

)

}νk
1{γj =1}(27)

is nondecreasing on 0 < νk ≤ 1.
Proof. First note that
( (m) )( (m) )( (m) )∑Vj
Vj
Vj
max exp≤
exp≤ #C̃k max exp
.
νk
νk
νk
j∈C̃k
j∈C̃k

(28)

j∈C̃k

Then we have


∑

(

(m)
max exp(Vj ) ≤
exp

j∈C̃k
j∈C̃k

(m)

Vj
νk

)νk



(

≤ #C̃k)νk(m)

max exp(Vj

).

(29)

j∈C̃k

Taking the limit as νk → 0, we obtain the desired result.
Next, we let 0 < νk < ν̃k ≤ 1. Then we haveνk
{( (m) )}νk
( (m) )∑VjVj
exp νkj∈C̃k exp
νk

∑


{
( (m) )}ν̃k = 
ν̃k 
(
{
)}
∑
(m)
νk 
Vj
j∈C̃k ∑V
exp jj∈C̃k exp
ν̃k
j∈C̃k



ν̃k


( (m) ) νν˜k νk

k
V

∑ 


exp jν̃k

 




( (m) )
=

∑


V

j∈C̃k   j∈C̃k exp jν̃k( (m) )



Vj
ν̃k

νk

exp

∑

( (m) ) 
≤

∑
Vj
j∈C̃k
exp
j∈C̃k
ν̃k
= 1.
Hence

{

∑

j∈Ck

(
exp

(30)
(m)

Vj
νk

is nondecreasing on 0 < νk ≤ 1.
23

)

}νk
1{γj =1}(31)

B

Proof of Theorem 1

In this appendix, we give the proof of Theorem 1. First, we show the following lemmaon the upper bound estimation on the conditional expectation on the number of rooms
booked.
Lemma 2.
(m)

E[RTi |NT= k] ≤ k  maxl∈Cki ,l̸=i 






1
1 + exp((m)(m) (m)
(xi )
Vl
−Vi

νki


)

{

1

+ max
l̸∈Cki1 + exp(Vl

(m)

(m)

− Vi

(m)

(xi ))(32)(m)

Proof. Noting that there exists j ̸= i such that qj= ∞, by (15), (16), we have(m)

E[RTi |NT = k]1k∑(m)γ(i(m)γ(i,...,i ),...,i )=ai(i1 ,...,ik ) pi1 1 k . . . pik 1 k(i1 ,...,ik )∈Sk

=

k
∑

∑

1
(m)γ(i,...,i )1{il =i} pi11kk(m)γ(i,...,i ). . . pik1kl=1 (i1 ,...,ik )∈Sk

=

k
∑

∑

1
(m)γ(i,...,il−1 ,i,il+1 ,...,ik )

1

pi1

l
(m)γ(i,...,i1. . . pil−1 ,i,il+1 ,...,ik )k(m)γ(i,...,i1. . . pi k

l−1 ,i,il+1 ,...,ik )

l=1 (i1 ,...,il−1 ,i,il+1 ,...,ik )∈Sk

≤

k
∑
l=1

∑

(m)γ

max piγ∈Γ,γj =1(i1 ,...,il−1 ,i,il+1 ,...,ik )∈Sk

1(m)γ(i
1 ,...,il−1 ,i,il+1 ,...,ik )

pi1

(m)γ

= k max pi
γ∈Γ,γj =1l−1(m)γ(i,...,i. . . pil−11

l−1 ,i,il+1 ,...,ik )

l+1(m)γ(i,...,i

pil+1

.

1

l−1 ,i,il+1 ,...,ik )k(m)γ(i,...,i. . . pik1l−1 ,i,il+1 ,...,ik )

(33)

24


}

.

Here we have used the equality∑(i1 ,...,il−1 ,i,il+1 ,...,ik )∈Sk
l−1(m)γ(i,...,i1(m)γ(i,...,ipi1=

l−1 ,i,il+1 ,...,ik )

1

L
∑

···

i1 =1

. . . pil−1

L
L
∑
∑

···

il−1 =1 il+1 =1

pi1
=

···

i1 =1

L
L
∑
∑

l−1 ,i,il+1 ,...,ik )

1

pil+1

k
(m)γ(i,...,il−1 ,i,il+1 ,...,ik )1. . . pikL∑l−1(m)γ(i,...,i. . . pil−1···

il−1 =1 il+1 =1

l−1 ,i,il+1 ,...,ik )1l+1(m)γ(i,...,il−1 ,i,il+1 ,...,ik )

1

pil+1

k
(m)γ(i,...,i. . . pik1l−1 ,i,il+1 ,...,ik )L∑
ik−1 =11(m)γ(i
1 ,...,il−1 ,i,il+1 ,...,ik )

pi1

l+1
(m)γ(i,...,iik =11(m)γ(i
1 ,...,il−1 ,i,il+1 ,...,ik )L∑

l−1 ,i,il+1 ,...,ik )1l−1(m)γ(i,...,i. . . pil−1

l−1 ,i,il+1 ,...,ik )1l+1(m)γ(i,...,il−1 ,i,il+1 ,...,ik )

1

pil+1

k−1
(m)γ(i,...,i. . . pik−1 1

l−1 ,i,il+1 ,...,ik )

...
= 1,

(34)

which follows from (3).
(m)γ(i) The case j ∈ Cki . In this case, since piin (8) is maximized at γ ∈ Γ such thatγi = γj = 1, γl = 0 (l ̸= i, j),V

maxγ∈Γ,γj =1(m)pγi = 1 ·

V

(m)

(m)

(xi
νki

exp( i

(m)

(xi
νki

exp( i
)

)

)
V

i

1

=

(m)

1 + exp(

Vj(m)) + exp( νjk )

(m)

(m)

−Vi
(xi
νki

)

.

(35)

)

(ii) The case j ̸∈ Cki . Let kj be the index in {1, . . . , n} such that j ∈ Ckj . In this case,for all γ ∈ Γ, γj = 1,
(m)γ

pi

{

∑

(m)

Vl
l∈Cki exp( νkiV)1{γl =1} }νki(m)exp( νi k )1{γi =1}

i
·∑
.
≤ ∑
(m)
(m)
(m)
V
Vl
V
exp(
)1
{ l∈Ck exp( νl k )1{γl =1} }νki + exp( νjk )νkj
{γl =1}
l∈Cki
νk
i

i

{

∑

i

(m)

V
)1{γl =1} }νki + exp( νjk
j

(m)

exp(

≤

(m)

exp(
=

Vi

Vi

(m)

(xi
νki

V

i

(m)

Vl
l∈Cki exp( νki

(m)

(xi
νki

) νk
i

)

i

j

When γl = 0 for all l ∈ Cki , l ̸= i,(m)∑V{ l∈Ck exp( νl k )1{γl =1} }νki
)νkj

(36)

·∑

(m)

exp( νi k )1{γi =1}
i

V

(m)

l
l∈Ck exp( νk )1{γl =1}
i

i

) νk
i

)

(m)

V
+ exp( νjkj1(m)(m) (m)1 + exp(Vj − Vi (xi ))

νkj

·1

)
.

(37)

25

When there exists l′ ∈ Cki , l′ ̸= i such that γl′ = 1,
{
{

∑

∑

(m)

Vl
l∈Cki exp( νki(m)Vll∈Ck exp( νk
i

i

exp(

≤1·

V

(m)

(x

(m)

V
)1{γl =1} }νki + exp( νjkj(m) (m)
Vi
(xi )

νki

(m)

exp( i νk i
 i

)

V

Hence
(m)γ

max piγ∈Γ,γj =1

(m)

i

(m)
(m) (m)
Vl
−Vi
(xi )

i


.

νki

(38)

)







≤ max  max l∈Cki ,l̸=iVll∈Ck exp( νk )1{γl =1}



1



i

(m)
i

1 + exp(

)νkj

·∑

(m)

exp( νi k )1{γi =1})) + exp( νl′k )≤ max l∈Cki ,l̸=iV)1{γl =1} }νki,

1
1 + exp((m)(m) (m)
Vl
−Vi
(xi )

νki



)



1

(m)
(m) (m) 
1 + exp(Vj − Vi (xi ))(39)
Thus we havemaxγ∈Γ,γj =1

pγi ≤





1

max

l∈Cki ,l̸=i 






V

(m)

1 + exp( l

(m)

(m)

−Vi
(xi
νki

)


)

{
+ max
l̸∈Cki}1
1 + exp(Vl

(m)

(m)

− Vi

(m)

(xi ))(40)

and the proof is complete.
Then we give the proof of Theorem 1.
Let
(m)
(m)
f (xi ) = xi

(m)

∞
∑

(m)

fN (xi ) = xi

k=0
N
∑

(λ(m) T )k
,
k!

(41)

(λ(m) T )k
.
k!

(42)

(m)

= k] exp(−λ(m) T )(m)= k] exp(−λ(m) T )E[RTi |NT

E[RTi |NT

k=0
(m)

(m)

Note that fN (xi ) is a continuous function on xi ∈ [0, ∞) by (15).
(m)(m)
First we show that fN (xi ) converges to f (xi ) uniformly on [0, ∞) as N → ∞. By

26.

Lemma 2,
(m)

(m)

xi

(m)

fN (xi ) − f (xi )sup∈[0,∞)∞∑(λ(m) T )kk!
(m)xi ∈[0,∞)k=N +1




}
{


∞
(m)
(m)


∑ 
xi
xi

k  max
≤ sup
+ max

(m)
(m) (m)(m)(m)(m)
l∈Cki ,l̸=i (m)
 1 + exp( Vl −Vi (xi ) )  l̸∈Cki 1 + exp(Vl − Vi (xi ))xi ∈[0,∞) k=N +1
=

sup

(m)

xi

(m)

E[RTi |NT= k] exp(−λ(m) T )νki(λ(m) T )kk!

× exp(−λ(m) T )∞∑≤M

k exp(−λ(m) T )k=N +1(λ(m) T )kk!

→ 0 (N → ∞),
where

(43)



M=

sup
(m)

xi

∈[0,∞)


 max





l∈Cki ,l̸=i 






(m)

xi
V

(m)

1 + exp( l

(m)

(m)

−Vi
(xi
νki

)


)

{

(m)

xi

+ max
l̸∈Cki1 + exp(Vl

(m)

(m)

− Vi

(m)

(xi ))}
 < ∞.

(44)
Here we have used the fact that since βi < 0,




{
}


(m)
(m)


xi
xi


+ max
lim  max
 = 0.
(m)(m) (m)
(m)
(m)
(m)
(m)
l∈Cki ,l̸=i xi →∞ l̸∈Cki 1 + exp(Vl − Vi (xi )) 1 + exp( Vl −Vi (xi ) ) 
νki

(45)
(m)

(m)

(m)

Hence fN (xi ) converges to f (xi ) uniformly on [0, ∞) and f (xi ) is a continuous(m)function on xi ∈ [0, ∞).

27

Similarly,(m)

0 ≤ f (xi )∞∑(λ(m) T )k
(m)
(m)
= xi
E[RTi |NT = k] exp(−λ(m) T )k!
 k=0 

}
{


(m)
(m)


xi
xi


≤  max+ max(m)(m) (m)(m)(m) (m)V−V(x)
l∈Cki ,l̸=i 
l̸
∈
C

ki
i
i
1 + exp(Vl − Vi (xi )) 1 + exp( l

)
νk
i

×

∞
∑
k=0

k exp(−λ(m) T )= λ(m) T  max

(λ

(m)





l∈Cki ,l̸=i 


k

T)
k!





(m)
xi
V

(m)

1 + exp( l

(m)

(m)

−Vi
(xi
νki

)


)

{
+ max
l̸∈Cki1 + exp(Vl

(m)
xi
(m)

(m)

− Vi

(m)

(xi ))


}

,

(46)
(m)

and it follows that limx(m) →∞ f (xi ) = 0.
i

(m)

(m)

Since f (xi ) is continuous, f (xi ) has a maximum point on any finite interval of theform [0, K], and for any suﬃciently small ϵ > 0, we can take K > 0 such that for any
(m)
(m)
(m)
(m)
xi > K, 0 ≤ f (xi ) < ϵ. Therefore f (xi ) has a maximum point in xi ∈ [0, ∞).

C

Proof of Lemma 1

Since the multinomial logit model is the special case of the nested logit model with
ν1 = · · · = νn = 1, we only prove (20).
First note that for j ∈ {1, . . . , L},( (m) )( (m))
Ṽj + α1 + δ1 y (m)Vjexp
1{γj =1} = exp
1{γj =1}νkνk( (m) ))((m)
Ṽj
α1 + δ1 y= exp
exp
1{γj =1} .
(47)
νk
νk
Therefore we have
(m)
(m)
∑
V
V
{ j∈Ck exp( νjk )1{γj =1} }νkiexp( νi k )1{γi =1}
(m)γ
i
i
i
=∑
pi
·∑
(m)
(m)
∑
V
Vj
n
j
νk
{
exp(
)1
}
exp(
)1{γj =1}
{γj =1}
k=1
j∈Ck
j∈Ck
νk
νk
exp(α1 + δ1 y (m) ){

=

∑

i

(m)

Ṽj
j∈Cki exp( νkii)1{γj =1} }νkiexp( α1 +δνk1 y

(m)

Ṽ

(m)

) exp( νi k )1{γi =1}

·
(m)
(m)
∑n ∑
(m) ∑
Ṽj
Ṽ
(m)
ν
k
exp(α1 + δ1 y ) k=1 { j∈Ck exp( νk )1{γj =1} }exp( α1 +δνk1 y ) j∈Ck exp( νjk )1{γj =1}
{

=∑

∑

i

i

(m)

Ṽj
j∈Cki exp( νki)1{γj =1} }νki

∑
Ṽj
n
k=1 {
j∈Ck exp( νk(m))1{γj =1}

}νk

i

i

i

(m)

·∑

28

Ṽ
exp( νi ki)1{γi =1}(m)

Ṽjj∈Cki exp( νki)1{γj =1}.

(48)